In what situations would you use a height/bump map?
A height map is useful if the camera is going to be close enough to the surface to "see through" the trickery of a normal map. A normal map leaves the surface where it is, and just changes how light reflects off; but if the "eye" is close enough that the difference in height itself would have been visible, that looks super weird.

Here's a heavily subdivided plane equipped with a normal map generated from a noise texture:



And here's the same plane, with the same texture, except that this time it's treated as a height map:



Isn't the second one a thousand times more believable? The first one would be good enough if seen from a great enough distance, or if the bump-map were "weakened" so that it reflected much shallower bumps; but this close up, only the height map can be realistic.